Interferometry is an art long used in optics to study light. The electromagnetic waves that make up a beam of light are commonly examined by directing the beam onto an object or surface and visually examining or photographically recording the patterns of reflected and scattered light from the object or surface. A beam of light from a sealed beam headlamp might, for example, be directed onto a screen, and the pattern of the illuminated area will reveal information about the filament, filament placement, and the quality of the parabolic reflector to one skilled in the art.
The wave nature of light has long been known, and it has long been known that light waves directed to the same position in space will "interfere". The principles of interference are simple: if two waves are directed toward the same point so that they overlap, the amplitude of the resulting wave at any instant will be the algebraic sum of the amplitudes of the two separate waves. Hence, if the two waves at an instant have the same absolute amplitude but one is positive and the other negative, the amplitudes will cancel, i.e. interfere, and there will be no light at that point at that instant. If the two waves are both positive or both negative in amplitude, the amplitudes will add, and the light intensity will be greater than for either wave alone. If the two waves considered have the same wavelength the result will be a standing wave.
A coherent beam of light, such as light produced by a laser, is light wherein all the waves have the same wavelength and their relative phases are stationary in time. Most naturally occurring and man-made light sources are not coherent, and interferometry is not very useful for characterizing substantially non-coherent beams, because little can be learned from the random interference patterns produced by such beams.
Coherent light is very desirable for many purposes in optics. For example, if one produces a lens for a telescope, and then uses the lens to focus a known coherent beam of light, one might qualify the lens by the aberrations caused to the beam by the lens as determined by viewing an interference pattern. Coherent light in an optics laboratory is very useful for many kinds of experiments and tests.
Another optical beam quality that is highly desirable for optical testing is collimation. A beam is said to be collimated when all rays or wave vectors of the light are exactly parallel. For light propagating in an isotropic medium, such as space or the atmosphere, the direction of propagation of the energy carried by the field, and therefore the wave vector of the field, is normal to the surface of constant optical phase (called the wavefront surface). Parallel wave vectors thus mean that the wavefront is flat and that the light is propagating in the form of a plane wave.
Light which is collimated as well as coherent is particularly useful in testing lenses, mirrors, and other optical elements. Another use of a collimated beam includes aligning and focusing optical systems such as telescopes. Also, collimated beams are produced for sending light over long distances, such as for laser communications, laser range finders, and laser designators, etc. These uses rely on the fact that collimated beams will expand the least over distance. Another example of the usefulness of collimation is in the use of lasers as cutting tools, such as in laser surgery which also requires the transportation of the laser beam over some distance from the laser to the focussing optic. A diverging beam would spread its energy over a broader area as it propagated through a medium such as air, and would soon lose sufficient energy density so as not to be able to perform its intended function. As another example, collimated beams are used to reduce wavefront curvature errors in interferometry, and to produce accurate reference beams in holography. Further, collimated beams are used to produce accurate and well characterized laser standing waves (counter-propagating plane waves) for nonlinear interaction experiments and for laser velocimetry and anemometry.
For many of the above uses, the lateral cross section of a beam as it exits a laser is often too small. Hence, it is common to expand the beam emitted from the laser by use of a beam-expanding telescope, which in simple form, is a tube with a lens at one end to cause the laser beam to diverge to a second lens at the other end, which then collimates the beam.
Once a laser beam is expanded and collimated, it is important to know the quality of the beam, such as the precision of collimation and the presence or absence of any aberrations in the wavefront. If the quality of the beam is not known, then for some applications the beam may not be particularly useful.
The most common test for collimation at the present time is to measure the beam diameter at two positions along the beam path. A perfectly collimated beam would show the smallest change in beam diameter along the beam path, so if the measured diameter is the same at two points, then the beam is approximately collimated. The problem with measuring the beam diameter at two positions is that it is a very inaccurate measurement. The reasons for the poor accuracy are first that the typical laser beam intensity profile has no sharp edges, but instead follows a Gaussian curve. It is difficult to judge with any accuracy exactly where the edge of the beam is. Second, all laser beams have spatial variation in the intensity caused by diffraction from dust particles in the beam path, dirty optics and other imperfections that cause changes in the intensity profile of the beam over relatively short propagation distances. These changes can substantially degrade the accuracy of measurement of the beam diameter.
Another approach for judging collimation is to use a lateral shearing plate interferometer. This instrument typically consists of an uncoated glass plate with flat surfaces. Some implementations have parallel surfaces, and some have a small angle (wedge) between the surfaces.
In a parallel plate shearing interferometer, the plate is placed at an angle relative to the incident beam to be characterized and two reflections of the incident beam result. One reflection is from the front surface of the glass plate and the other is from the back surface. (Other multiple reflections also occur between the surfaces before they exit the front surface. These multiply reflected beams are, however, much lower in intensity than the two principal beams described, and hence will be ignored in the ensuing analysis.) A screen or other plane, opaque, surface placed in the path of the reflected beams will show the optical axes of two beams separated by a distance s, called lateral shear, which is a function of thickness of the plate, the index of refraction of the shearing plate material, and the angle of incidence.
FIG. 1A shows a plan view arrangement of a lateral shearing plate interferometer with a laser source 111 producing a beam 113 incident on a first end of a beam expanding telescope 114. Beam 113 continues from the other end of the beam expanding telescope, approximately collimated but slightly diverging, and is incident on a shearing plate 115. The beam incident on the shearing plate is shown diverging to an exaggerated degree to facilitate illustration of virtual sources. Distance d between the lenses (not shown) of the beam expanding telescope is adjustable to collimate the expanded beam. (The refraction within the lenses is not shown for ease of illustration.) The outer edges of the incident beam reflect from the front surface of the shearing plate at points 117 and 119.
The outer edges of the incident beam reflect from the back surface at points 121 and 123. The reflected beams, partially superimposed, travel in the direction of arrow 125 away from the shearing plate, and appear to come from two different virtual sources, one at point 127 and the other at point 129. These two virtual sources have a lateral separation, s, the lateral shear, and also an axial separation, 1, called the axial delay. The details of the geometry can be seen more clearly from FIG. 1B. For a shearing interferometer with index of refraction n and thickness t, the shear of the two sources produced by reflection of a beam incident at angle theta is given by EQU s=2 d.sub.1 cos.theta.,
where d.sub.1 is shown in FIG. 1B. From Snell's Law, EQU sin.theta.=nd.sub.1 /(t.sup.2 +d.sub.1.sup.2).sup.1/2
Solving for d.sub.1 EQU d.sub.1 =t sin.theta./(n.sup.2 -sin.sup.2 .theta.).sup.1/2.
Thus the shear between the two reflected beams is EQU s=t sin 2.theta./(n.sup.2 -sin.sup.2 .theta.).sup.1/2.
Similarly, the axial delay 1, is given by EQU 1=2nd.sub.2 -d.sub.3
where EQU d.sub.2 =nd.sub.1 /sin.theta.
and EQU d.sub.3 =2d.sub.1 sin.theta..
Hence, the axial delay is EQU 1=2t (n.sup.2 -sin.sup.2 .theta.).sup.1/2. (1)
FIG. 2 shows a theoretical superposition of the two reflected beams of FIG. 1A on a plane surface 131 of FIG. 1A viewed in the direction of arrow 125. The beam scattered from the front surface is represented by circle 133 on surface 131 in FIG. 2 and the beam reflected from the back surface is represented by circle 135. The two beam circles are offset by the distance of the lateral shear s. Area 137 is the area of superposition and interference of the reflected beams. If the laser beam is reasonably free of aberrations and close to collimation, the interference area will show parallel alternating light and dark bands (fringes), approximately as shown in FIG. 2. (To first order, the fringe pattern is sinusoidal in intensity.)
A good treatment of lateral shearing interferometry can be found in the book Optical Shop Testing, published by John Wiley and Sons, Inc. of New York, N.Y., and edited by Daniel Malacara, pages 108 through 141, incorporated herein by reference. As may be seen in the referenced publication, interferometry may be used to divine considerable information about an incident beam and any apparatus that is used to emit, transmit, or manipulate the beam.
In the arrangement of FIG. 1A, the spatial period D1 of the vertical fringes is substantially a function of the wavelength Lambda of the beam, the radius of curvature R of the incident wavefront, and the lateral shear s: EQU D1=(Lambda.times.R)/s
As a user adjusts the focus of the beam-expanding telescope by changing d, the spacing of the lenses, the collimation of the beam is changed, thereby changing the radius of curvature R of the propagating wavefront. The radius of curvature can be positive or negative, depending on whether the beam is diverging (convex) or converging (concave), respectively. If the focus of the telescope is changed so the magnitude of the radius of curvature increases, i.e. the wavefront becomes flatter, D1 increases. R increases as the point of wavefront flatness is approached, and R=infinity for a flat wavefront. D1, directly proportional to R, increases as well, and at R=infinity, D1 is also infinite. At some point approaching collimation, depending on Lambda and s, D1 is greater than the width of the region 137 and the user will no longer see vertical fringes. The superimposed region will appear to have uniform brightness, either dark or light.
This presents an essential problem when using the technique: namely that perfect collimation is not necessarily achieved when the fringe pattern disappears, because all that is really known is that D1 has become larger than the width of the overlapping area. If one continues to move the focus in the same direction until the fringes reappear, and then finds a midpoint between the position at where the fringes first disappeared and the position at where they reappear, that will be the point of approximate collimation. There is, however, in the arrangement described above, no precise way to find the midpoint. An additional problem with this device is that there is no indication of the direction to focus the telescope until d is changed.
To improve the accuracy of the lateral shearing plate interferometer, plates have been produced with a small angle (wedge) between the flat surfaces. Using a wedged shearing plate, as the position of collimation is approached, the fringe pattern rotates from the vertical and becomes horizontal. In a wedged shearing plate interferometer it is common to provide a thin wire extending across the plate to cast a horizontal shadow on the screen for a standard for alignment of the interference fringes.
A problem with a wedged shearing plate interferometer is that precise alignment with a shadow line is often difficult. Moreover, an accurate refrence line is difficult to achieve for the standard at the screen, and the elements of the arrangement, such as the interferometer and the screen, must be precisely positioned relative to one another for the shadow line at the screen to represent precise beam collimation. In addition, the angle of the fringes in the fringe pattern change rather slowly, rather than dramatically, as R passes through infinity, so there can be a resolution problem as well.
What is needed is an instrument for determining collimation that is relatively insensitive to alignment, is easy to use, provides a sensitive and precise indication of wavefront flatness, and indicates to a user the direction required when moving the collimating telescope in order to achieve collimation. Such an apparatus should not sacrifice utility for indicating wavefront aberrations, and should avoid the added complexity incurred in interpreting the fringe pattern from a wedged shearing plate interferometer.